\subsection{Comment from Shizhong}
I talked with Shizhong last night.  He suggested me to work with eq \eqref{eq:kSch}, introduce a new equation and do either renomalization on $v$ directly, or by do some kind of cross integrate like Tony did in his note.  For further effect, I can introduce a form factor $f(p)$ into the integration.  For a clear Fermi see in normal state, it is a step function and for BEC/resonance/BCS state, it is smeared.  But either case has $f(\infty)\rightarrow1$, and this might be enough for renormalization.  All these might show some clue how to do the many-body instead of one close-channel molecule.  